## MINI-COURSES

### Carolyn ABBOTT (Brandeis University)

#### Hyperbolically embedded subgroups of acylindrically hyperbolic groups

## Abstract

This course is an introduction to acylindrically hyperbolic groups through the lens of the special class of hyperbolically embedded subgroups. We will begin with definitions and examples, and then we will investigate some nice geometric (and algebraic) properties enjoyed by these subgroups. We will explore connections with random walks, extending actions, extending quasicocycles, and relative hyperbolicity. Hyperbolically embedded subgroups, particularly those that are virtually cyclic, are a powerful tool for proving important results about acylindrically hyperbolic groups. I will introduce some of these results and sketch the ways in which hyperbolically embeddded subgroups are used in their proof. The course will assume no prior knowledge of any of these topics.

### Kasia JANKIEWICZ (University of California Santa Cruz)

#### Bestvina-Brady Morse theory and its applications

## Abstract

This mini-course will be an introduction to the Bestvina-Brady Morse theory. Let G be a group acting geometrically on a polyhedral complex X. Suppose that φ:G -> Z is an epimorphism, and X admits a φ-equivarient map to a line. The Bestvina-Brady Morse theory relates the finiteness properties of the kernel of φ to topological properties of the link of a vertex in X. We will focus on the applications of the theory in the context of right-angled Artin and Coxeter groups.

### Robert KROPHOLLER (University of Warwick)

#### l^2 homology of groups and consequences

## Abstract

After a brief introduction to the general theory of homology of groups we will define l^2 homology via actions on simplicial complexes. In the second lecture we will compute several examples of these homology groups. Finally, we will discuss consequences that are of interest to group theorists particularly in the case that these invariants vanish.

### Jason MANNING (Cornell University)

#### Quasiconvexity in (relatively) hyperbolic groups

## Abstract

I’ll talk about various ways to think about quasiconvexity in hyperbolic and relatively hyperbolic groups, and explain why and when quasiconvexity persists under Dehn filling.

### Jean Pierre MUTANGUHA (Princeton University)

#### Dynamics to geometry in free-by-cyclic groups

## Abstract

Given an automorphism of a finitely generate free group, one can construct a corresponding semi-direct product of the free group and the integers. The semi-direct product, also known as a free-by-cyclic group, is finitely generated and we can consider its Cayley graph (or some other geometric model you prefer). We are interested in the dynamics of the defining automorphism, the algebraic properties of the free-by-cyclic group, the coarse geometry of the geometric models, and especially the interplay between these. These ideas are heavily influenced by Thurston’s work on surface homeomorphisms and 3-manifolds. This mini-course will give an incomplete survey of the various results (from old to recent) along these lines and introduce some of the key ideas behind them. There will be no shortage of open problems to take home with you.

## RESEARCH TALKS

### Macarena ARENAS (University of Cambridge)

#### Classifying spaces for quotients of cubulated groups

## Résumé

We’ll explore the problem of finding ‘nice’ models for the classifying spaces of certain quotients of fundamental groups of non-positively curved cube complexes, and we’ll discuss the framework that provides the necessary tools to do so – cubical small-cancellation theory.

### Matt CLAY (University of Arkansas)

#### Bounded projections in the Z−factor graph

## Abstract

Let GGG be a free product G=A1∗A2∗⋯∗Ak∗FNG = A_1 * A_2* \cdots * A_k * F_NG=A1∗A2∗⋯∗Ak∗FN, where each of the groups AiA_iAi is torsion-free and where FNF_NFN is a free group of rank NNN, and let O\mathcal{O}O be the associated deformation space. We show that the diameter of the projection of the subset of O\mathcal{O}O where a given element has bounded length to the Z\mathcal{Z}Z–factor graph is bounded, where the diameter bound depends only on the length bound. This relies on an analysis of the boundary of GGG as a hyperbolic group relative to the collection of subgroups AiA_iAi together with a given non-peripheral cyclic subgroup. In a future paper, we will apply this theorem to study the geometry of free group extensions. This is joint work with Caglar Uyanik.

### François DAHMANI (Université Grenoble Alpes)

#### A Minkowski separation for certain subgroups of mapping tori of free groups

## Abstract

A group G satisfies the Minkowski separation if if has a finite quotient Q in which all finite order outer-automorphisms of G descend as automorphisms of Q and are non trivial in Out(Q). This property was identified as useful in order to compare two relatively hyperbolic groups whose JSJ decomposition is given. Applied to certain subgroups of mapping tori (F \rtimes Z) of a free group F, it allows to compare the conjugacy classes of certain automorphisms of F. In this talk, the groups of interest are the subgroups of suspensions by multi-Dehn twists. (Joint work with N. Touikan.)

### Pallavi DANI (Louisiana State University)

#### Subgroup distortion in hyperbolic groups

## Abstract

The distortion function of a subgroup measures the extent to which the intrinsic word metric of the subgroup differs from the metric induced by the ambient group. Ol’shanskii showed that there are almost no restrictions on which functions arise as distortion functions of subgroups of finitely presented groups. This prompts one to ask what happens if one forces the ambient group to be particularly nice, say, for example, to be hyperbolic. I will survey which functions are known to be distortion functions of subgroups of hyperbolic groups and describe joint work with Tim Riley in which we construct new examples of such functions.

### Jonah B. GASTER (University of Wisconsin-Milwaukee)

#### The Markov ordering of the rationals

## Abstract

A rational number p/q determines a simple closed curve on a once-punctured torus. When the torus is endowed with a complete hyperbolic metric, each rational gets a well-defined length. If the metric is chosen so that the torus is “modular” (that is, when its holonomy group is conjugate into PSL(2,Z)), the lengths of the curves have special arithmetic significance, with connections to Diophantine approximation and number theory. Taking inspiration from McShane’s elegant proof of Aigner’s conjectures, concerning the (partial) ordering of the rationals induced by hyperbolic length on the modular torus, I will describe how hyperbolic geometry can be used to characterize monotonicity of the order so obtained along lines of varying slope in the (q,p)-plane.

### Daniel GROVES (University of Illinois at Chicago)

#### Relatively geometric actions on cube complexes

## Abstract

Einstein and I introduced relatively geometric actions of relatively hyperbolic groups and instigated the study of relatively geometric actions on CAT(0) cube complexes. I will introduce these actions, give examples and non-examples, and explain how one gets separability of certain quasi-convex subgroups under appropriate hypotheses. I’ll compare the theory of relatively geometric actions to that of geometric actions of hyperbolic groups. This is mostly joint work with Teddy Einstein, and some joint work with Corey Bregman and Kejia Zhu.

### Chris HRUSKA (University of Wisconsin-Milwaukee)

#### Local connectivity of the boundary of a relatively hyperbolic group

## Abstract

One of the first major results about Gromov hyperbolic groups states that the boundary of any one-ended hyperbolic group is always locally connected. This deep result—established in the 1990s by Bestvina-Mess, Levitt, Bowditch, and Swarup—led to advances in JSJ decompositions, boundary classification problems, and the analysis of boundaries. Shortly afterward, Bowditch introduced a natural boundary for any relatively hyperbolic group pair and conjectured that this boundary is always locally connected. However, he established this fact only in the presence of certain awkward and restrictive hypotheses.

In fact, the boundary of every one-ended relatively hyperbolic group pair is locally connected, with no restrictions. This result applies in a general setting, in which the groups in question need not be finitely generated or even countable. (Joint work with Ashani Dasgupta.)

### Nir LAZAROVICH (Technion Institute)

#### Finite index rigidity of hyperbolic groups

## Abstract

We show that isomorphic finite index subgroups of non-elementary hyperbolic groups must have the same index. In this talk I will present the tools and ideas of the proof.

### Marco LINTON (University of Oxford)

#### The coherence of one-relator groups

## Abstract

(Joint work with Andrei Jaikin-Zapirain.) A group is said to be coherent if all of its finitely generated subgroups are finitely presented. In this talk I will sketch a proof of Baumslag’s well known conjecture that all one-relator groups are coherent and discuss applications of the ingredients that go into the proof.

### Rylee LYMAN (Rutgers University–Newark)

#### Outer spaces of virtually free groups

## Abstract

A group is virtually free when it has a free subgroup of finite index. A theorem of Karrass, Pietrowski and Solitar says that equivalently a group is finitely generated and virtually free when it acts on a tree with finite stabilizers and finite quotient. We will introduce the deformation space of these trees, called Outer Space, and relate it to the Outer Space